Theorem padicabvcxp 25321

Description: All positive powers of the p-adic absolute value are absolute values. (Contributed by Mario Carneiro, 9-Sep-2014.)

Hypotheses

Ref	Expression
qrng.q	|- Q = (ℂfld ↾s QQ )
qabsabv.a	|- A = (AbsVal ` Q )
padic.j	|- J = ( q e. Prime |-> ( x e. QQ |-> if ( x = 0 , 0 , ( q ^ -u ( q pCnt x ) ) ) ) )

Assertion

Ref	Expression
padicabvcxp	|- ( ( P e. Prime /\ R e. RR+ ) -> ( y e. QQ |-> ( ( ( J ` P ) ` y ) ^c R ) ) e. A )

Distinct variable groups:   x, q, y   y, J   A, q, x, y   x, Q, y   P, q, x, y   R, q, y

Allowed substitution hints:   Q( q)   R( x)   J( x, q)

Proof of Theorem padicabvcxp

Step	Hyp	Ref	Expression
1		padic.j	. . . . . . 7 |- J = ( q e. Prime |-> ( x e. QQ |-> if ( x = 0 , 0 , ( q ^ -u ( q pCnt x ) ) ) ) )
2	1	padicval 25306	. . . . . 6 |- ( ( P e. Prime /\ y e. QQ ) -> ( ( J ` P ) ` y ) = if ( y = 0 , 0 , ( P ^ -u ( P pCnt y ) ) ) )
3	2	adantlr 751	. . . . 5 |- ( ( ( P e. Prime /\ R e. RR+ ) /\ y e. QQ ) -> ( ( J ` P ) ` y ) = if ( y = 0 , 0 , ( P ^ -u ( P pCnt y ) ) ) )
4	3	oveq1d 6665	. . . 4 |- ( ( ( P e. Prime /\ R e. RR+ ) /\ y e. QQ ) -> ( ( ( J ` P ) ` y ) ^c R ) = ( if ( y = 0 , 0 , ( P ^ -u ( P pCnt y ) ) ) ^c R ) )
5		ovif 6737	. . . . 5 |- ( if ( y = 0 , 0 , ( P ^ -u ( P pCnt y ) ) ) ^c R ) = if ( y = 0 , ( 0 ^c R ) , ( ( P ^ -u ( P pCnt y ) ) ^c R ) )
6		rpre 11839	. . . . . . . . . 10 |- ( R e. RR+ -> R e. RR )
7	6	adantl 482	. . . . . . . . 9 |- ( ( P e. Prime /\ R e. RR+ ) -> R e. RR )
8	7	recnd 10068	. . . . . . . 8 |- ( ( P e. Prime /\ R e. RR+ ) -> R e. CC )
9		rpne0 11848	. . . . . . . . 9 |- ( R e. RR+ -> R =/= 0 )
10	9	adantl 482	. . . . . . . 8 |- ( ( P e. Prime /\ R e. RR+ ) -> R =/= 0 )
11	8, 10	0cxpd 24456	. . . . . . 7 |- ( ( P e. Prime /\ R e. RR+ ) -> ( 0 ^c R ) = 0 )
12	11	adantr 481	. . . . . 6 |- ( ( ( P e. Prime /\ R e. RR+ ) /\ y e. QQ ) -> ( 0 ^c R ) = 0 )
13	12	ifeq1d 4104	. . . . 5 |- ( ( ( P e. Prime /\ R e. RR+ ) /\ y e. QQ ) -> if ( y = 0 , ( 0 ^c R ) , ( ( P ^ -u ( P pCnt y ) ) ^c R ) ) = if ( y = 0 , 0 , ( ( P ^ -u ( P pCnt y ) ) ^c R ) ) )
14	5, 13	syl5eq 2668	. . . 4 |- ( ( ( P e. Prime /\ R e. RR+ ) /\ y e. QQ ) -> ( if ( y = 0 , 0 , ( P ^ -u ( P pCnt y ) ) ) ^c R ) = if ( y = 0 , 0 , ( ( P ^ -u ( P pCnt y ) ) ^c R ) ) )
15		df-ne 2795	. . . . . 6 |- ( y =/= 0 <-> -. y = 0 )
16		pcqcl 15561	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ ( y e. QQ /\ y =/= 0 ) ) -> ( P pCnt y ) e. ZZ )
17	16	adantlr 751	. . . . . . . . . . . . 13 |- ( ( ( P e. Prime /\ R e. RR+ ) /\ ( y e. QQ /\ y =/= 0 ) ) -> ( P pCnt y ) e. ZZ )
18	17	zcnd 11483	. . . . . . . . . . . 12 |- ( ( ( P e. Prime /\ R e. RR+ ) /\ ( y e. QQ /\ y =/= 0 ) ) -> ( P pCnt y ) e. CC )
19	8	adantr 481	. . . . . . . . . . . 12 |- ( ( ( P e. Prime /\ R e. RR+ ) /\ ( y e. QQ /\ y =/= 0 ) ) -> R e. CC )
20		mulneg12 10468	. . . . . . . . . . . 12 |- ( ( ( P pCnt y ) e. CC /\ R e. CC ) -> ( -u ( P pCnt y ) x. R ) = ( ( P pCnt y ) x. -u R ) )
21	18, 19, 20	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ R e. RR+ ) /\ ( y e. QQ /\ y =/= 0 ) ) -> ( -u ( P pCnt y ) x. R ) = ( ( P pCnt y ) x. -u R ) )
22	19	negcld 10379	. . . . . . . . . . . 12 |- ( ( ( P e. Prime /\ R e. RR+ ) /\ ( y e. QQ /\ y =/= 0 ) ) -> -u R e. CC )
23	18, 22	mulcomd 10061	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ R e. RR+ ) /\ ( y e. QQ /\ y =/= 0 ) ) -> ( ( P pCnt y ) x. -u R ) = ( -u R x. ( P pCnt y ) ) )
24	21, 23	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ R e. RR+ ) /\ ( y e. QQ /\ y =/= 0 ) ) -> ( -u ( P pCnt y ) x. R ) = ( -u R x. ( P pCnt y ) ) )
25	24	oveq2d 6666	. . . . . . . . 9 |- ( ( ( P e. Prime /\ R e. RR+ ) /\ ( y e. QQ /\ y =/= 0 ) ) -> ( P ^c ( -u ( P pCnt y ) x. R ) ) = ( P ^c ( -u R x. ( P pCnt y ) ) ) )
26		prmuz2 15408	. . . . . . . . . . . . . . 15 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
27	26	adantr 481	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ R e. RR+ ) -> P e. ( ZZ>= ` 2 ) )
28		eluz2b2 11761	. . . . . . . . . . . . . 14 |- ( P e. ( ZZ>= ` 2 ) <-> ( P e. NN /\ 1 < P ) )
29	27, 28	sylib 208	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ R e. RR+ ) -> ( P e. NN /\ 1 < P ) )
30	29	simpld 475	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ R e. RR+ ) -> P e. NN )
31	30	nnrpd 11870	. . . . . . . . . . 11 |- ( ( P e. Prime /\ R e. RR+ ) -> P e. RR+ )
32	31	adantr 481	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ R e. RR+ ) /\ ( y e. QQ /\ y =/= 0 ) ) -> P e. RR+ )
33	17	znegcld 11484	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ R e. RR+ ) /\ ( y e. QQ /\ y =/= 0 ) ) -> -u ( P pCnt y ) e. ZZ )
34	33	zred 11482	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ R e. RR+ ) /\ ( y e. QQ /\ y =/= 0 ) ) -> -u ( P pCnt y ) e. RR )
35	32, 34, 19	cxpmuld 24480	. . . . . . . . 9 |- ( ( ( P e. Prime /\ R e. RR+ ) /\ ( y e. QQ /\ y =/= 0 ) ) -> ( P ^c ( -u ( P pCnt y ) x. R ) ) = ( ( P ^c -u ( P pCnt y ) ) ^c R ) )
36	7	renegcld 10457	. . . . . . . . . . 11 |- ( ( P e. Prime /\ R e. RR+ ) -> -u R e. RR )
37	36	adantr 481	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ R e. RR+ ) /\ ( y e. QQ /\ y =/= 0 ) ) -> -u R e. RR )
38	32, 37, 18	cxpmuld 24480	. . . . . . . . 9 |- ( ( ( P e. Prime /\ R e. RR+ ) /\ ( y e. QQ /\ y =/= 0 ) ) -> ( P ^c ( -u R x. ( P pCnt y ) ) ) = ( ( P ^c -u R ) ^c ( P pCnt y ) ) )
39	25, 35, 38	3eqtr3d 2664	. . . . . . . 8 |- ( ( ( P e. Prime /\ R e. RR+ ) /\ ( y e. QQ /\ y =/= 0 ) ) -> ( ( P ^c -u ( P pCnt y ) ) ^c R ) = ( ( P ^c -u R ) ^c ( P pCnt y ) ) )
40	30	nnred 11035	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ R e. RR+ ) -> P e. RR )
41	40	recnd 10068	. . . . . . . . . . 11 |- ( ( P e. Prime /\ R e. RR+ ) -> P e. CC )
42	41	adantr 481	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ R e. RR+ ) /\ ( y e. QQ /\ y =/= 0 ) ) -> P e. CC )
43	30	nnne0d 11065	. . . . . . . . . . 11 |- ( ( P e. Prime /\ R e. RR+ ) -> P =/= 0 )
44	43	adantr 481	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ R e. RR+ ) /\ ( y e. QQ /\ y =/= 0 ) ) -> P =/= 0 )
45	42, 44, 33	cxpexpzd 24457	. . . . . . . . 9 |- ( ( ( P e. Prime /\ R e. RR+ ) /\ ( y e. QQ /\ y =/= 0 ) ) -> ( P ^c -u ( P pCnt y ) ) = ( P ^ -u ( P pCnt y ) ) )
46	45	oveq1d 6665	. . . . . . . 8 |- ( ( ( P e. Prime /\ R e. RR+ ) /\ ( y e. QQ /\ y =/= 0 ) ) -> ( ( P ^c -u ( P pCnt y ) ) ^c R ) = ( ( P ^ -u ( P pCnt y ) ) ^c R ) )
47	31, 36	rpcxpcld 24476	. . . . . . . . . . 11 |- ( ( P e. Prime /\ R e. RR+ ) -> ( P ^c -u R ) e. RR+ )
48	47	adantr 481	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ R e. RR+ ) /\ ( y e. QQ /\ y =/= 0 ) ) -> ( P ^c -u R ) e. RR+ )
49	48	rpcnd 11874	. . . . . . . . 9 |- ( ( ( P e. Prime /\ R e. RR+ ) /\ ( y e. QQ /\ y =/= 0 ) ) -> ( P ^c -u R ) e. CC )
50		rpne0 11848	. . . . . . . . . 10 |- ( ( P ^c -u R ) e. RR+ -> ( P ^c -u R ) =/= 0 )
51	48, 50	syl 17	. . . . . . . . 9 |- ( ( ( P e. Prime /\ R e. RR+ ) /\ ( y e. QQ /\ y =/= 0 ) ) -> ( P ^c -u R ) =/= 0 )
52	49, 51, 17	cxpexpzd 24457	. . . . . . . 8 |- ( ( ( P e. Prime /\ R e. RR+ ) /\ ( y e. QQ /\ y =/= 0 ) ) -> ( ( P ^c -u R ) ^c ( P pCnt y ) ) = ( ( P ^c -u R ) ^ ( P pCnt y ) ) )
53	39, 46, 52	3eqtr3d 2664	. . . . . . 7 |- ( ( ( P e. Prime /\ R e. RR+ ) /\ ( y e. QQ /\ y =/= 0 ) ) -> ( ( P ^ -u ( P pCnt y ) ) ^c R ) = ( ( P ^c -u R ) ^ ( P pCnt y ) ) )
54	53	anassrs 680	. . . . . 6 |- ( ( ( ( P e. Prime /\ R e. RR+ ) /\ y e. QQ ) /\ y =/= 0 ) -> ( ( P ^ -u ( P pCnt y ) ) ^c R ) = ( ( P ^c -u R ) ^ ( P pCnt y ) ) )
55	15, 54	sylan2br 493	. . . . 5 |- ( ( ( ( P e. Prime /\ R e. RR+ ) /\ y e. QQ ) /\ -. y = 0 ) -> ( ( P ^ -u ( P pCnt y ) ) ^c R ) = ( ( P ^c -u R ) ^ ( P pCnt y ) ) )
56	55	ifeq2da 4117	. . . 4 |- ( ( ( P e. Prime /\ R e. RR+ ) /\ y e. QQ ) -> if ( y = 0 , 0 , ( ( P ^ -u ( P pCnt y ) ) ^c R ) ) = if ( y = 0 , 0 , ( ( P ^c -u R ) ^ ( P pCnt y ) ) ) )
57	4, 14, 56	3eqtrd 2660	. . 3 |- ( ( ( P e. Prime /\ R e. RR+ ) /\ y e. QQ ) -> ( ( ( J ` P ) ` y ) ^c R ) = if ( y = 0 , 0 , ( ( P ^c -u R ) ^ ( P pCnt y ) ) ) )
58	57	mpteq2dva 4744	. 2 |- ( ( P e. Prime /\ R e. RR+ ) -> ( y e. QQ |-> ( ( ( J ` P ) ` y ) ^c R ) ) = ( y e. QQ |-> if ( y = 0 , 0 , ( ( P ^c -u R ) ^ ( P pCnt y ) ) ) ) )
59		rpre 11839	. . . . 5 |- ( ( P ^c -u R ) e. RR+ -> ( P ^c -u R ) e. RR )
60	47, 59	syl 17	. . . 4 |- ( ( P e. Prime /\ R e. RR+ ) -> ( P ^c -u R ) e. RR )
61		rpgt0 11844	. . . . 5 |- ( ( P ^c -u R ) e. RR+ -> 0 < ( P ^c -u R ) )
62	47, 61	syl 17	. . . 4 |- ( ( P e. Prime /\ R e. RR+ ) -> 0 < ( P ^c -u R ) )
63		rpgt0 11844	. . . . . . . 8 |- ( R e. RR+ -> 0 < R )
64	63	adantl 482	. . . . . . 7 |- ( ( P e. Prime /\ R e. RR+ ) -> 0 < R )
65	7	lt0neg2d 10598	. . . . . . 7 |- ( ( P e. Prime /\ R e. RR+ ) -> ( 0 < R <-> -u R < 0 ) )
66	64, 65	mpbid 222	. . . . . 6 |- ( ( P e. Prime /\ R e. RR+ ) -> -u R < 0 )
67	29	simprd 479	. . . . . . 7 |- ( ( P e. Prime /\ R e. RR+ ) -> 1 < P )
68		0red 10041	. . . . . . 7 |- ( ( P e. Prime /\ R e. RR+ ) -> 0 e. RR )
69	40, 67, 36, 68	cxpltd 24465	. . . . . 6 |- ( ( P e. Prime /\ R e. RR+ ) -> ( -u R < 0 <-> ( P ^c -u R ) < ( P ^c 0 ) ) )
70	66, 69	mpbid 222	. . . . 5 |- ( ( P e. Prime /\ R e. RR+ ) -> ( P ^c -u R ) < ( P ^c 0 ) )
71	41	cxp0d 24451	. . . . 5 |- ( ( P e. Prime /\ R e. RR+ ) -> ( P ^c 0 ) = 1 )
72	70, 71	breqtrd 4679	. . . 4 |- ( ( P e. Prime /\ R e. RR+ ) -> ( P ^c -u R ) < 1 )
73		0xr 10086	. . . . 5 |- 0 e. RR*
74		1re 10039	. . . . . 6 |- 1 e. RR
75	74	rexri 10097	. . . . 5 |- 1 e. RR*
76		elioo2 12216	. . . . 5 |- ( ( 0 e. RR* /\ 1 e. RR* ) -> ( ( P ^c -u R ) e. ( 0 (,) 1 ) <-> ( ( P ^c -u R ) e. RR /\ 0 < ( P ^c -u R ) /\ ( P ^c -u R ) < 1 ) ) )
77	73, 75, 76	mp2an 708	. . . 4 |- ( ( P ^c -u R ) e. ( 0 (,) 1 ) <-> ( ( P ^c -u R ) e. RR /\ 0 < ( P ^c -u R ) /\ ( P ^c -u R ) < 1 ) )
78	60, 62, 72, 77	syl3anbrc 1246	. . 3 |- ( ( P e. Prime /\ R e. RR+ ) -> ( P ^c -u R ) e. ( 0 (,) 1 ) )
79		qrng.q	. . . 4 |- Q = (ℂfld ↾s QQ )
80		qabsabv.a	. . . 4 |- A = (AbsVal ` Q )
81		eqid 2622	. . . 4 |- ( y e. QQ |-> if ( y = 0 , 0 , ( ( P ^c -u R ) ^ ( P pCnt y ) ) ) ) = ( y e. QQ |-> if ( y = 0 , 0 , ( ( P ^c -u R ) ^ ( P pCnt y ) ) ) )
82	79, 80, 81	padicabv 25319	. . 3 |- ( ( P e. Prime /\ ( P ^c -u R ) e. ( 0 (,) 1 ) ) -> ( y e. QQ |-> if ( y = 0 , 0 , ( ( P ^c -u R ) ^ ( P pCnt y ) ) ) ) e. A )
83	78, 82	syldan 487	. 2 |- ( ( P e. Prime /\ R e. RR+ ) -> ( y e. QQ |-> if ( y = 0 , 0 , ( ( P ^c -u R ) ^ ( P pCnt y ) ) ) ) e. A )
84	58, 83	eqeltrd 2701	1 |- ( ( P e. Prime /\ R e. RR+ ) -> ( y e. QQ |-> ( ( ( J ` P ) ` y ) ^c R ) ) e. A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   RR*cxr 10073   < clt 10074   -ucneg 10267   NNcn 11020   2c2 11070   ZZcz 11377   ZZ>=cuz 11687   QQcq 11788   RR+crp 11832   (,)cioo 12175   ^cexp 12860   Primecprime 15385   pCnt cpc 15541   ↾_s cress 15858  AbsValcabv 18816  ℂ_fldccnfld 19746   ^c ccxp 24302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-dvds 14984  df-gcd 15217  df-prm 15386  df-pc 15542  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-mulg 17541  df-subg 17591  df-cntz 17750  df-cmn 18195  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-drng 18749  df-subrg 18778  df-abv 18817  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-cxp 24304

This theorem is referenced by:  ostth3  25327  ostth  25328